32edo: Difference between revisions

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== Octave stretch or compression ==

What follows is a comparison of compressed-octave 32edo tunings.

; 32edo

* Step size: 37.500{{c}}, octave size: 1200.0{{c}}

Pure-octaves 32edo approximates all harmonics up to 16 within 15.5{{c}}.

{{Harmonics in equal|32|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 32edo}}

{{Harmonics in equal|32|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 32edo (continued)}}

; [[WE|32et, 13-limit WE tuning]]

* Step size: 37.481{{c}}, octave size: 1199.4{{c}}

Compressing the octave of 32edo by around half a cent results in improved primes 3, 7 and 11, but worse primes 5 and 13. This approximates all harmonics up to 16 within 18.3{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.

{{Harmonics in cet|37.481|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 32et, 13-limit WE tuning}}

{{Harmonics in cet|37.481|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 32et, 13-limit WE tuning (continued)}}

; [[WE|32et, 11-limit WE tuning]]

* Step size: 37.453{{c}}, octave size: 1198.5{{c}}

Compressing the octave of 32edo by around 1.5{{c}} results in improved primes 3, 7 and 11, but worse primes 2, 5 and 13. This approximates all harmonics up to 16 within 16.4{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this.

{{Harmonics in cet|37.453|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 32et, 11-limit WE tuning}}

{{Harmonics in cet|37.453|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 32et, 11-limit WE tuning (continued)}}

; [[ed7|90ed7]]

* Step size: 37.431{{c}}, octave size: 1197.8{{c}}

Compressing the octave of 32edo by around 2{{c}} results in improved primes 3, 7, 11 and 13, but worse primes 2 and 5. This approximates all harmonics up to 16 within 18.6{{c}}. If one wishes to use both of 32edo's mappings of the 5th harmonic simultaneously, this tuning is suited to that due to evenly sharing the error between them. The tuning 90ed7 does this.

{{Harmonics in equal|90|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 90ed7}}

{{Harmonics in equal|90|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 90ed7 (continued)}}

; [[zpi|133zpi]]

* Step size: 37.418{{c}}, octave size: 1197.375{{c}}

Compressing the octave of 32edo by around NNN{{c}} results in improved primes 3, 7, 11 and 13, but worse primes 2 and 5. This approximates all harmonics up to 16 within 17.4{{c}}. The tuning 133zpi does this.

{{Harmonics in cet|37.418|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 133zpi}}

{{Harmonics in cet|37.418|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 133zpi (continued)}}

Below is a plot of the [[Zeta]] function, showing how its peak (ie biggest absolute value) is shifted above 32, corresponding to a zeta tuning with octaves flattened to 1197.375 cents. This will improve the fifth, at the expense of the third.

[[File:plot32.png|alt=plot32.png|plot32.png]]

; [[51edt]]

* Step size: 37.293{{c}}, octave size: 1193.4{{c}}

Compressing the octave of 32edo by around 6.5{{c}} results in improved primes 3, 5 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 18.2{{c}}. The tuning 51edt does this.

{{Harmonics in equal|51|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 51edt}}

{{Harmonics in equal|51|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 51edt (continued)}}

== Instruments ==

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+== Octave stretch or compression ==
+What follows is a comparison of compressed-octave 32edo tunings.
+; 32edo
+* Step size: 37.500{{c}}, octave size: 1200.0{{c}}
+Pure-octaves 32edo approximates all harmonics up to 16 within 15.5{{c}}.
+{{Harmonics in equal|32|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 32edo}}
+{{Harmonics in equal|32|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 32edo (continued)}}
+; [[WE|32et, 13-limit WE tuning]]
+* Step size: 37.481{{c}}, octave size: 1199.4{{c}}
+Compressing the octave of 32edo by around half a cent results in improved primes 3, 7 and 11, but worse primes 5 and 13. This approximates all harmonics up to 16 within 18.3{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
+{{Harmonics in cet|37.481|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 32et, 13-limit WE tuning}}
+{{Harmonics in cet|37.481|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 32et, 13-limit WE tuning (continued)}}
+; [[WE|32et, 11-limit WE tuning]]
+* Step size: 37.453{{c}}, octave size: 1198.5{{c}}
+Compressing the octave of 32edo by around 1.5{{c}} results in improved primes 3, 7 and 11, but worse primes 2, 5 and 13. This approximates all harmonics up to 16 within 16.4{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this.
+{{Harmonics in cet|37.453|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 32et, 11-limit WE tuning}}
+{{Harmonics in cet|37.453|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 32et, 11-limit WE tuning (continued)}}
+; [[ed7|90ed7]]
+* Step size: 37.431{{c}}, octave size: 1197.8{{c}}
+Compressing the octave of 32edo by around 2{{c}} results in improved primes 3, 7, 11 and 13, but worse primes 2 and 5. This approximates all harmonics up to 16 within 18.6{{c}}. If one wishes to use both of 32edo's mappings of the 5th harmonic simultaneously, this tuning is suited to that due to evenly sharing the error between them. The tuning 90ed7 does this.
+{{Harmonics in equal|90|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 90ed7}}
+{{Harmonics in equal|90|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 90ed7 (continued)}}
+; [[zpi|133zpi]]
+* Step size: 37.418{{c}}, octave size: 1197.375{{c}}
+Compressing the octave of 32edo by around NNN{{c}} results in improved primes 3, 7, 11 and 13, but worse primes 2 and 5. This approximates all harmonics up to 16 within 17.4{{c}}. The tuning 133zpi does this.
+{{Harmonics in cet|37.418|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 133zpi}}
+{{Harmonics in cet|37.418|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 133zpi (continued)}}
+Below is a plot of the [[Zeta]] function, showing how its peak (ie biggest absolute value) is shifted above 32, corresponding to a zeta tuning with octaves flattened to 1197.375 cents. This will improve the fifth, at the expense of the third.
+[[File:plot32.png|alt=plot32.png|plot32.png]]
+; [[51edt]]
+* Step size: 37.293{{c}}, octave size: 1193.4{{c}}
+Compressing the octave of 32edo by around 6.5{{c}} results in improved primes 3, 5 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 18.2{{c}}. The tuning 51edt does this.
+{{Harmonics in equal|51|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 51edt}}
+{{Harmonics in equal|51|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 51edt (continued)}}
 == Instruments ==